Study on Certain Subclass of Analytic Functions Involving Mittag- Leffler Function

Department of Mathematical Sciences, Faculty of Science, Princess Nourah Bint Abdulrahman University, Saudi Arabia Department of Mathematics, GSS, GITAM University, Doddaballapur, 562 163, Bengaluru Rural, Karnataka, India Research and Development Wing, Live4Research, Tiruppur, 638 106 Tamilnadu, India Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, Rajasthan, India Department of Mathematics, Kakatiya University, Warangal, 506 009 Telangana, India Department of Humanities and Management, Jigme Namgyel Engineering College, Royal University of Bhutan, Dewathang, Bhutan


Introduction
Let A represent the regular function class u defined on the disk U = fw : jwj < 1g normalized by u (i.e., uð0Þ = 0 and u ′ ð0Þ = 1). The origin of the form is about the Taylor series expansion of such an equation S indicates a subclass of A consists entirely of mappings that are the same as U.
For u ∈ A presented by (2) and gðwÞ specified by their convolution, represented by ðu * gÞ, is specified as u * g ð Þ w ð Þ = w + 〠 ∞ n=2 a n b n w n = g * u ð Þ w ð Þ w ∈ U ð Þ: ð3Þ The A subclass consisting of the u-type function is specified by T as Silverman [1] extensively examined this subclass. The study of operators plays an important role in geometric function theory in complex analysis and its related fields. Many derivative and integral operators can be written in terms of convolution of certain analytic functions. It is observed that this formalism brings an ease in further mathematical exploration and also helps to better understand the geometric properties of such operators. The Mittag-Leffler function [2,3] is defined by the following power series, convergent in the whole complex plane: We recognize that it is an entire function of order 1/υ providing a simple generalization of the exponential function expðwÞ to which it reduces for υ = 1: For detailed information on the Mittag-Leffler-type functions and their laplace transforms, the reader may consult, e.g., [4][5][6] and the recent treatise by Gorenflo et al. [7].
We also note that for the convergence of the power series in (5), the parameter υ may be complex provided that RðυÞ > 0: The most interesting properties of the Mittag-Leffler function are associated with its asymptotic expansions as w ⟶ ∞ in various sectors of the complex plane. A more general function E υ,τ generalizing E υ ðwÞ was introduced by Wiman [8] and defined by Observe that the function E υ,τ contains many wellknown functions as its special case, for example, E 1,1 ðwÞ = e w , E 1,2 ðwÞ = ðe w − 1Þ/w, E 2,1 ðw 2 Þ = cosh w, E 2,1 ð− w 2 Þ = cos w, E 2,2 ðw 2 Þ = sinh w/w, E 2,2 ð−w 2 Þ = sinh w/w, E 4 = 1/2½cos w 1/4 + cosh w 1/4 , and E 3 = 1/2½e w 1/3 + 2e −ð1/3Þ cos ðð ffiffi ffi 3 p /2Þw 1/3 Þ: The Mittag-Leffler function arises naturally in the solution of fractional-order differential and integral equations and especially in the investigations of fractional generalization of kinetic equation, random walks, Levy flights, and super diffusive transport and in the study of complex systems. Several properties of Mittag-Leffler function and generalized Mittag-Leffler function can be found, e.g., in [9][10][11][12][13][14][15][16]. Observe that Mittag-Leffler function E υ,τ ðwÞ does not belong to the family A: Thus, it is natural to consider the following normalization of Mittag-Leffler functions as below: It holds for complex parameters υ, τ and w ∈ ℂ: The function Q υ,τ ðwÞ is specified by Now, for u ∈ A, the derivative operator that follows is If u is specified by (1), then from the operator's definition D m ℏ u, it is clear to see that where Keep in mind that (1) the Al-Oboudi operator [17] is achieved when υ = 0 and τ = 1 (2) we get the Salagean operator [18] when υ = 0, τ = 1, and ℏ = 1 (3) when m = 0, we get E υ,τ ðwÞ, according to Srivastava et al. [19] If u ∈ T is represented by (4), then we have got it.
Now, by utilizing the differential operator, D m ℏ ðυ, τÞu, a new subclass of functions belonging to the class A is specified.
The objective of this review is to look into a variety of properties for functions in the aforesaid class. For specific parameter instances.

2
Journal of Function Spaces

Coefficient Estimates
To get our results, we will require the subsequent lemma.
For beginnings, we have a coefficient that is relevant for functions in the classs S m ℏ,υ,τ ðν, ℓ, k,℘Þ: where then u ∈ S m ℏ,υ,τ ðν, ℓ, k,℘Þ: Proof. In the definition by consequence of 1 and Lemma 2, it is enough to demonstrate that For the R.H.S and L.H.S of (18), we may, respectively, write and similarly, Then, The condition (16) required is fulfilled.
We have a necessary and adequate situation in the next theorem for a function u ∈ T to be in the class T S m ℏ,υ,τ ðν, ℓ, k,℘Þ.

Journal of Function Spaces
Equations (25) and (26) are sharp for the u given function Proof. Since u ∈ TS m ℏ,υ,τ ðν, ℓ, k,℘Þ and it follows from 4 of the theorem, where B η ðℏ, ν, m, υ, τ, ℓ, k,℘Þ is given by (27), we have and therefore, Since u is given by (3), we get In light of Theorem 4, we have which yields Thus, Hence, the proof is complete.

Extreme Points
Now, for the function class, we look at the extreme points TS m ℏ,υ,τ ðν, ℓ, k,℘Þ.
The evidence 9 and 10 of the subsequent theorems is comparable to 8 of the theorem, so the evidence is excluded.

Fekete-Szego Inequality
In this section, for the mapping in the class, we get the Fekete-Szego inequality S m ℏ,υ,τ ðν, ℓ, k,℘Þ. To illustrate our fundamental result, we will identify the appropriate lemma.
Lemma 11 (see [27]). If pðwÞ = 1 + c 1 w + c 2 w + c 3 w 2 + ⋯ is an analytic mapping with positive real part in U, then When ȷ < 0 or ȷ > 1, the inequality holds iff pðwÞ = ð1 + wÞ/ð1 − wÞ or one of its rotations. If 0 < ȷ < 1, then the equality holds iff or one of its rotations. If ȷ = 0, the equality holds iff or one of its rotations: If ȷ = 1, the equality holds iff pðwÞ is the reciprocal of one of the mapping such that the equality holds when it comes to ȷ = 0: The outcome is sharp.
Proof. Since, for complex numbers, RðzÞ ≤ jzj, u ∈ S m ℏ,υ,τ ðν, ℓ, k,℘Þ implies that or that Hence, Let We then have, by way of (10) and (14), Therefore, we obtain We write where The implementation of the lemma above follows our conclusion. Denote If μ < σ 1 or μ > σ 2 , it is true that equality exists.
When σ 1 < μ < σ 2 , it is true that equality exists, iff Journal of Function Spaces If μ = σ 1 , then it is true that equality exists, iff Finally, if it is true that equality exists ⟺pðwÞ μ = σ2, it is the inverse of one of the equality functions and holds true in the case of μ = σ2

Partial Sums
Consider the recent works on partial analytic function sums by Silverman [28] and Silvia [29]. Partial function in this class is considered in this section to be TS m ℏ,υ,τ ðν, ℓ, k,℘Þ giving sharp lower boundaries to the reap part ratios of uðwÞ to uqðwÞ and u′ðwÞ to u′qðwÞ: Theorem 13. Let u ∈ TS m ℏ,υ,τ ðν, ℓ, k,℘Þ and indicate u 1 ðwÞ and u q ðwÞ as partial sums Suppose that where Then, u ∈ TS m ℏ,υ,τ ðν, ℓ, k,℘Þ: Furthermore, Proof. It is not crucial to verify that the d η coefficients supplied by (69) are correct.
The hypothesis used (69), by setting If we use and apply (73), we find that That immediately leads in a conclusion (70) of Theorem 13. To find out that gives sharp result, we observe that for w = re iπ/q , Similarly, if we take we can deduce, and make use of (73), that This leads directly to the statement (71) of Theorem 13. For each q ∈ ℕ with the external mapping uðwÞ, the bound in (71) is sharp indicated by (76).
Thus, the evidence of the Theorem 13 is complete.

Journal of Function Spaces
Proof. By setting Now, Now, since the L.H.S. of (83) is bounded above by and the proof is complete.

Conclusions
This research has introduced study a new differential operator related to analytic function and studied some basic properties of geometric function theory. Accordingly, some results to coefficient estimates, grouth and distortion theorem, Fekete-Szego inequalityy, and partial sums have also been considered, inviting future research for this field of study.

Data Availability
No data were used to find this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.